For an invertible matrix A, prove that A and A-1 have the same eigenvectors. How are the eigenvalues of A related to the eigenvalues of A-¹? Letting x be an eigenvector of A gives Ax = λx for a corresponding eigenvalue λ. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = λx AXA-1 = AXA-1 0x44−1 = 24-1 xl = λA-¹x x = λ4-1x A-1x = 1x 2 Ax = λx A-¹Ax= A-¹2x Ix = 2A-¹x x = λ4-1x 1x 2 A-¹x = This shows that Need Help? 1/x Ax = x Ax/A = λx/A (A/A)X = 2x4-1 Ix = AXA-1 x = AXA-1 A-¹x = 1x λ 1/A Ax = λχ A/(Ax) = A/(2x) (A/A)x= (A/2)x Ix = (A/A)x x = 2A-¹x -Select-- is an eigenvector of A-¹ with eigenvalue -Select- A ¹x = 1x 2 -Select---
For an invertible matrix A, prove that A and A-1 have the same eigenvectors. How are the eigenvalues of A related to the eigenvalues of A-¹? Letting x be an eigenvector of A gives Ax = λx for a corresponding eigenvalue λ. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = λx AXA-1 = AXA-1 0x44−1 = 24-1 xl = λA-¹x x = λ4-1x A-1x = 1x 2 Ax = λx A-¹Ax= A-¹2x Ix = 2A-¹x x = λ4-1x 1x 2 A-¹x = This shows that Need Help? 1/x Ax = x Ax/A = λx/A (A/A)X = 2x4-1 Ix = AXA-1 x = AXA-1 A-¹x = 1x λ 1/A Ax = λχ A/(Ax) = A/(2x) (A/A)x= (A/2)x Ix = (A/A)x x = 2A-¹x -Select-- is an eigenvector of A-¹ with eigenvalue -Select- A ¹x = 1x 2 -Select---
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:For an invertible matrix A, prove that A and A-1 have the same eigenvectors. How are the eigenvalues of A related to the eigenvalues of A-¹?
Letting x be an eigenvector of A gives Ax = 2x for a corresponding eigenvalue λ. Using matrix operations and the properties of inverse matrices gives which of the following?
Ax = x
Ax = Ax
Ax/A = x/A
(A/A)X = 2xA-1
Ix = AXA-1
AXA-1 = 2xA-1
OXAA-¹ = 24-¹x
XI = λA-¹x
x = 2A-¹x
A-1x = 1x
λ
Ax = λκ
A-¹Ax = A-¹1x
Ix = A=¹x
x = λ4-1x
A-1x = 1x
λ
O
This shows that
X
x = 2xA-1
X
Need Help? 1/x
λ
1/2
A-¹x = 1x
2
Ax = 2x
A/(Ax) = A/(2x)
(A/A)X = (A/2)x
Ix = (A/A)x
x = 2A-¹x
A¹x = 1x
2
-Select-- is an eigenvector of A-¹ with eigenvalue-Select-
-Select-
Transcribed Image Text:that A and A-1 have the same eigenvectors. How are the
gives Ax = λx for a corresponding eigenvalue λ. Using mat
Ax
λx
Ax
A = 2x/A
AxA-1 -
x =
λx4-1
x =
λx4-1
X =
λxA-1
-
X =
Ax = λx
A/(Ax) = A/(2x)
O(A/A)x (A/2)x
(A/2)x
24-1x
Ix =
X =
A-¹x = 1x
2
n eigenvector of A-1 with eigenvalue
1X
λ
-
OXAA-1
1/x
2
1/2
Select--
--Select-
wwwwwwEK
XI
X
A=1x
А
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